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Acta Mechhttps://doi.org/10.1007/s00707-017-2002-5
ORIGINAL PAPER
J. Ravnik · C. Marchioli · A. Soldati
Application limits of Jeffery’s theory for elongated particletorques in turbulence: a DNS assessmentThis paper is dedicated to the memory of Franz Ziegler
Received: 18 March 2017 / Revised: 20 July 2017© Springer-Verlag GmbH Austria 2017
Abstract Non-spherical particles suspended in fluid flows are subject to hydrodynamic torques generatedby fluid velocity gradients. For small axisymmetric particles, the most popular formulation of hydrodynamictorques is that given by Jeffery (Proc R Soc Lond A 102:161–179, 1922), which is valid for uniform shearflow in the viscous Stokes regime. In the lack of simple alternative formulations outside the Stokes regime, theJeffery formulation has been widely applied to inertial particles in turbulent flows, where it is bound to produceinaccurate results. In this paper we quantify the statistical error incurred when the Jeffery formulation is usedto study the motion of elongated axisymmetric particles under nonlinear shear flow conditions. Consideringthe archetypical case of prolate ellipsoidal particles in turbulent channel flow, we show that error for ellipsoidsof the same length, l, as the Kolmogorov scale of the flow, ηK , is indeed small (order 1%) but increasesexponentially up to l � 10ηK before becoming almost independent of elongation.
1 Introduction
Suspensions of non-spherical particles in turbulent flow are commonly found in a broad variety of industrialapplications (e.g., pulp and paper making, colloids and polymer manufacturing, post-combustion soot emis-sions) and environmental processes (e.g., atmospheric pollen dispersion, plankton and marine snow dynamicsin water bodies, ice cloud formation). These suspensions are characterized by complex particles–fluid interac-tions that depend strongly on particle shape and orientation. In recent years, a growing number of numerical[7,9–11,14,21,28,35,46–48] and experimental studies [8,15,17,26,31,33,36,42] have investigated such inter-actions, considering in particular the case of spheroidal particles translating and rotating at low concentrationsin a turbulent flow (see [43] for a review). Limiting our discussion to numerical studies, analyses have reliedon approaches that are routinely adopted to simulate turbulent dispersed flows [2]. The most detailed approachis the particle-resolved approach, which solves for the flow around each particle prescribing exact boundaryconditions for forces and torques at the particle surface. Computational costs associated with the calcula-tion of these boundary conditions, however, restrict current application of this approach to relatively small
J. RavnikFaculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, SloveniaE-mail: [email protected]
C. Marchioli (B) · A. SoldatiDepartment of Engineering and Architecture, University of Udine, Via delle Scienze 206, 33100 Udine, ItalyE-mail: [email protected]
A. SoldatiInstitute of Fluid Mechanics and Heat Transfer, TU Wien, Getreidemarkt 9, 1060 Wien, AustriaE-mail: [email protected]
J. Ravnik et al.
Fig. 1 Snapshots of one rigid fiber translating and rotating in turbulent channel flow (simulation parameters: shear Reynoldsnumber Reτ = 150; fiber Stokes number St = 25; fiber aspect ratio λ = 50—See Sect. 2 for definitions). Colors from blue to redin the background represent increasingly high values of the streamwise fluid velocity. The green solid line shows the trajectoryof the center of mass of the fiber. The time lag between each panel is 3 in wall units (color figure online)
numbers of particles [19]. Hence, the most applied approach to simulate large swarms of particles is thepoint-particle approach, in which particles are tracked individually, determining their translational dynamicsby Newton’s second law and their rotational dynamics by conservation of angular momentum for a rigid body.This Lagrangian approach is valid in the limit of particle size much smaller than the Kolmogorov lengthscale and has been widely used to investigate the motion of small spheroidal particles in simple shear flows[11,16,39] as well as turbulent flow [14,20–22,24,25,44,48]. One qualitative example of application is pro-vided in Fig. 1, where the motion of a prolate spheroid (mimicking a rigid fiber) in the near-wall region of aturbulent channel flow is shown. The actual length of the fiber is magnified to highlight its rotational motion,mostly characterized by tumbling in the mean-flow-gradient plane. Such motion results from the applicationof a lumped-parameter model that is valid in the limit of dilute flow conditions (one-way coupling). Albeitsimplified, the model incorporates the dependence of the hydrodynamic viscous drag force on the particleorientation, which is known to create crucial differences between elongated fibers and equivalent sphericalparticles. This has proven sufficient to capture the key physics coupling particle orientation to the velocity gra-dients and coupling particle translational motion to particle rotation (see [43] for a detailed review). The largemajority of numerical studies is based on such a model, which computes the hydrodynamic resistance forces(drag, in particular) and torques acting on the particles using expressions derived for creeping flow conditions[4,5,13,18]: Under these conditions, the trajectory of individual particles can be calculated using only the
Application limits of Jeffery’s theory
flow velocities and velocity gradients at the position of the particle’s center of mass. The first formulation wasproposed by [18] for the case of a single inertia-free ellipsoid in uniform shear flow under Stokes conditions.This formulation was later revisited by Bretherthon [6], who added the aspect ratio of the ellipsoid as shapeparameter to incorporate the effect of particle elongation. Further theoretical work has led to extensions of theJeffery equations for forces and torques to arbitrary flow fields (e.g., [4,5]) and to non-axisymmetric ellipsoids(e.g., [16]).
An important assumption, inherent in these formulations, is that the Reynolds number of the particle, Rep,is very small (ideally equal to zero): Away from this limit, inaccurate estimations of forces and torques shouldbe expected. In most real-life situations, however, particles are characterized by a finite value of Rep [2,43,46]because the particle inertia is high enough to generate a non-negligible particle-to-fluid velocity differenceand/or at least one particle dimension is large relative to the Kolmogorov scale. In this latter case, changes inthe fluid velocity gradients along the particle may occur due to nonlinear variations of the velocity profile. Suchrestrictions seem to pose strong limits to the applicability of the Jeffery equations to situations of practicalinterest, particularly for the calculation of hydrodynamic torques in turbulent flow conditions: The sensitivity ofrotational dynamics to errors associated with the velocity gradients is significantly higher than the sensitivityof translation dynamics to errors associated with the velocity [22,46]. Nevertheless, these equations havebeen verified experimentally [1,11,27,30,31,40,41], even for particles longer than the Kolmogorov scale [29].They have also been employed in many studies to investigate the behavior of anisotropic particles in laminarand turbulent flows [7,9,14,21,22,30,38,48], even outside the small-Rep limit (see [21]; Zhao et al. [9];Zhao et al. [22] among others). In this context, an important open issue is the level of inaccuracy associatedwith the use of the Jeffery formulation to calculate the hydrodynamic torques at small, yet finite Rep. Upto now, no quantification of such inaccuracy has been provided and a clear assessment of the applicationlimits of the Jeffery formulation is missing. In this paper, we provide this assessment for the reference caseof elongated prolate ellipsoids (mimicking a suspension of rigid fibers) in fully developed turbulent channelflow. Particles with different inertia (Stokes numbers from 1 to 25) and size (length l ranging from 0.1 to 36time the Kolmogorov length scale ηK ) are considered. The analysis is based on statistics of the error incurredwhen the Jeffery torques are used in the angular conservation momentum equations under nonlinear shear flowconditions, exploiting a direct numerical simulation database. Our objective is to determine an upper boundfor the ratio l/ηK above which the error becomes too high to be tolerated.
2 Governing equations
The carrier fluid is Newtonian (with dynamic viscosity μ and kinematic viscosity ν) and incompressible (withdensity ρ). The fluid motion is governed by the following dimensionless mass and momentum conservationequations, in vector form:
∇ · u = 0; ∂u∂t
+ (u · ∇)u = −∇ p + 1
Reτ
∇2u, (1)
where u and p denote the fluid velocity and the pressure, respectively. The frictional Reynolds number isReτ = huτ /ν, with h the channel half-height, the wall-friction velocity uτ=
√τwall/ρ is the wall-friction
velocity based on the wall shear stress τwall. The flow is driven by a constant mean pressure gradient and isunaffected by the presence of the fibers (one-way coupling).
Particles are modeled as pointwise rigid prolate spheroids with semi-major axis b, semi-minor axis a andaspect ratio λ = b/a. Particle translation and rotation are governed by the following equations:
mpdup
dt= FD, (2)
d(I·ω′)dt
+ ω′ × (I·ω′) = M′, (3)
where ρp is particle density, andmp = 4πa3λρp/3 is particlemass. In Eq. (2), up = dxp/dt is the translationalparticle velocity andFD is the drag force acting on the particle, both formulated in the inertial frame of referencex = 〈x, y, z〉 with x , y and z the streamwise, spanwise and wall-normal flow directions, respectively. In Eq.(3), I is the moment of inertia tensor, ω′ is the angular velocity of the particle andM′ is the torque. Both ω′ andM′ are formulated in the particle frame of reference x′ = 〈x ′, y′, z′〉 with origin at the fiber center of mass and
J. Ravnik et al.
coordinate axes x ′, y′ and z′ aligned with the principal directions of inertia. In the limit of pointwise particles,the surrounding flow can be considered as Stokesian and the drag force FD can be expressed as [4,5]:
FD = μRK′RT · (u f − up) = μK · u, (4)
where R is the orthogonal transformation matrix which relates the same vector in the two above-mentionedframes through the linear transformation x = R x′ (RT being its transpose), u f is the fluid velocity at the fiberposition, K′ is the resistance tensor in the particle frame [13,44], and u = u f − up is the relative velocitybetween the fluid and the particle at the center of mass of the particle (referred to as slip velocity hereinafter).Equation (4) is valid for a particle with arbitrary shape under creeping flow conditions, namely small fiberReynolds number, Rep = 2a | u | /ν.
The Jeffery torques divided by the moment of inertia of the ellipsoid are given as [18]
⎧⎪⎨
⎪⎩
MxJ /Ix ′
MyJ /Iy′
MzJ/Iz′
⎫⎪⎬
⎪⎭= 20
(a+)2
ρ f
ρp
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
1(β0+λ2γ0)
[1−λ2
1+λ2f ′ + (ξ ′ − ωx ′)
]
1(α0+λ2γ0)
[λ2−11+λ2
g′ + (η′ − ωy′)]
1(α0+β0)
(χ ′ − ωz′)
⎫⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎭
, (5)
where ρ f is the fluid density, f ′, g′ are elements of the deformation rate tensor, and ξ ′, η′ and χ ′ are elementsof the spin tensor:
f ′ = 1
2
(∂uz′
∂y′ + ∂uy′
∂z′
)
, g′ = 1
2
(∂ux ′
∂z′+ ∂uz′
∂x ′
)
, (6)
ξ ′ = 1
2
(∂uz′
∂y′ − ∂uy′
∂z′
)
, η′ = 1
2
(∂ux ′
∂z′− ∂uz′
∂x ′
)
, χ ′ = 1
2
(∂uy′
∂x ′ − ∂ux ′
∂y′
)
. (7)
The non-dimensional coefficients α0, β0 and γ0 are given as [13]:
α0 = β0 = λ2
λ2 − 1+ λ
2(λ2 − 1)3/2ln
λ − √λ2 − 1
λ + √λ2 − 1
, (8)
λ2γ0 = − 2λ2
λ2 − 1− λ3
(λ2 − 1)3/2ln
λ − √λ2 − 1
λ + √λ2 − 1
. (9)
Equation (2) can be recast as
dup
dt= K · (u f − up)
Stf (λ), (10)
where the particle Stokes number is defined as [37]:
St = 4
3
ρp
ρ f(a+)2λ f (λ), f (λ) = ln(λ + √
λ2 − 1)
6√
λ2 − 1, (11)
where the superscript + indicates wall units, obtained using the friction velocity uτ based on the mean wallshear stress and the fluid kinematic viscosity, ν (a+ = auτ /ν).
3 Simulation details
The statistics presented in this paper are relative to a turbulent Poiseuille channel flow of an incompressibleNewtonian fluid at shear Reynolds number Reτ = uτh/ν = 150. Here, uτ is the friction velocity based on themean wall shear stress and on fluid density, ν is the fluid kinematic viscosity and h is the channel half-height.The corresponding bulk Reynolds number is Re = u0h/ν = 2250, where u0 = 1.77 m/s is the bulk velocity.Direct numerical simulation of turbulence was performed using a pseudo-spectral flow solver [20,21]. Periodicboundary conditions were imposed in the streamwise (x) and spanwise (y) directions while no-slip boundary
Application limits of Jeffery’s theory
Table 1 Summary of simulation parameters for the particles
a+ λ l+η+K
ρpρ f
∣∣∣St=1
ρpρ f
∣∣∣St=5
ρpρ f
∣∣∣St=25
0.036 3 0.108 4276.19 − −0.12 3 0.36 34.72 173.61 868.060.36 3 1.08 42.76 213.81 1069.050.36 10 3.6 26.58 132.88 664.420.72 10 7.2 6.64 33.22 166.10.36 50 18 17.36 86.79 433.950.72 50 36 4.34 21.7 108.49
The smallest value of the Kolmogorov length scale in the channel, η+K = 2, is used to normalize particle length l+ = 2a+λ. The
shortest particles were not simulated at St = 5 and 25 because the corresponding density, yield by Eq. (11), would have been toohigh and not representative of any known material
conditions were imposed at the two walls. Time integration was performed using a second-order Adams–Bashforth scheme for the nonlinear terms, which are calculated upon de-aliasing in the periodic directions,and an implicit Crank–Nicolson scheme for the viscous terms. The domain size is 4πh × 2π h × 2 h in thestreamwise, spanwise and wall-normal directions, respectively. This domain is discretized by 128×128×129grid nodes.
Lagrangian tracking was performed considering particles with Stokes number St = 1, 5, 25 and aspectratio λ = 3, 10, 50. The simulation parameters for each set of particles are summarized in Table 1. Notethat, in the present flow configuration, the Kolmogorov scale ranges from a minimum value η+
K ,min � 2 (in
wall units) close to the wall to a maximum η+K ,max � 4 in the center of the channel [32]: In Table 1, the
dimensionless particle length l+ = 2a+λ, in wall units, is normalized by η+K = η+
K ,min. The same assumptionis made in the following section to provide conservative estimates of the application limits of the Jefferyformulation for torques. For each combination of values for St and λ (with the exception of λ = 3 andSt = 5, 25, which would have yielded unphysically high values of the particle density), Np = 105 particleswere injected into the flow with random orientation and velocity equal to the fluid velocity at the particlerelease position. The particle motion equations were solved using a 4th order Runge–Kutta scheme with atime step dt+ = 0.1 in wall units, equal to one-tenth of the smaller Stokes number considered. Such step sizeis small enough to ensure accurate time integration of the particle motion equations [12]. Periodic boundaryconditions are imposed on particles in both streamwise and spanwise directions, whereas elastic reflectionis applied when either of the particle’s ends touches the wall. Elastic reflection was chosen since it is themost conservative assumption when studying the reasons of particle preferential concentration in a turbulentboundary layer. The numerical implementation is based on the work of Ravnik and Hribersek [34]. Particlestatistics were computed starting at time t+ = 300 upon particle injection: This time span is long enough toensure that the initial conditions have no influence of the statistics. Statistics were then gathered for a time spant+ = 1000.
4 Results
As mentioned in Introduction, the Jeffery formulation for torques is based on the assumption that the fluidvelocity field varies linearly along the particle. In this case, the formulation is accurate and the velocity gradientsat the particle center of mass can be used to compute the particle rotational dynamics [18]. In a turbulent flow,this condition is met when particles are smaller than the Kolmogorov scale. However, recent experimentalresults on the rotation rate of rigid rod-like particles in homogeneous isotropic turbulence [30,38] have shownthat, compared to inertialess tracers, the particle rotation rate variance does not change up to particle lengthsof about 7ηK . This suggests that the Jeffery formulation might be applicable to particles as long as (or evenlonger than) the Kolmogorov eddies, with tolerable accuracy in the estimation of the torques. The objective ofthis section is precisely to quantify this accuracy and assess the application limits of Eq. (5): This is done bycharacterizing statistically the error associated with the use of the velocity gradients at the particle center ofmass.
J. Ravnik et al.
-1 -0.5 0 0.5 1-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.11.083.61836
2zl
∂u∂z
l
lηk
Fig. 2 Profiles of the instantaneous velocity gradient ∂u/∂z along the particle at varying particle length. The values of ∂u/∂zrefer to the locations shown in the top-right inset. To emphasize the change in ∂u/∂z a particle oriented perpendicularly to themean flow direction and located in the channel center is considered
4.1 Assessment of the linear-velocity assumption
We examine first the behavior of the flow field along the particle length l. To this aim, we sampled thefluid velocity gradients at eleven locations uniformly distributed along l for each particle in each set. Theselocations are indicated in the top-right inset of Fig. 2, which shows the behavior of one reference velocitygradient component, ∂u/∂z: This specific component was chosen because it is associated with the mainshear component in the flow. The profiles correspond to different values of l+/η+
K and refer to a particlethat is instantaneously oriented in the cross-sectional plane with respect to the mean flow direction: For thisparticular orientation, ∂u/∂z is expected to exhibit the largest spatial variation. As could be anticipated, thevelocity gradient is indeed uniform for the short particles (l+/η+
K = 0.1), meaning that the velocity fieldvaries linearly and can thus be approximated using a Taylor series expansion around the particle center, asdone by Jeffery to derive Eq. (5). However, the profile remains nearly flat also for l+/η+
K = 1.08, and smoothvariations are observed only starting from l+/η+
K = 3.6. As the particle length increases further, changes inthe velocity gradient become more and more evident, with rather large oscillations for l+/η+
K ≥ 18: For theseparticles, which are likely to span several small-scale turbulent structures at a time, the surrounding velocityflow field varies nonlinearly.We remark here that the behavior of the velocity gradients depends strongly on theorientation of the particle relative to themean flow.When the particle is oriented in the streamwise direction, weobserve much smaller variations for all gradient components (results not shown): In this situation, applicationof the Jeffery formulation beyond the Stokes regimemay still provide accurate estimates of the hydrodynamicstorques. The observation is particularly relevant for the case of inertial particles accumulating in the near-wallregion of the channel, which are known to preferentially align in the mean flow direction, namely the directionof strongest Lagrangian stretching [45]. More so because orientation plays an important role particularly forshort particles, while having weaker effects on longer particles: Short particles (e.g., sub-Kolmogorov) interactwith a single small-scale flow structure at a time and the fluid velocity variation they “see” changes dependingon their orientation; on the other hand, long particles (e.g., spanning a few Kolmogorov length scales) may besubject to the action of different flow structures and the variation of fluid velocities “seen” has weak statisticaldependence on orientation.
To characterize the variation of the fluid velocities and velocity gradients along the particle, we considertwo norms. The first norm is defined as
||uJf − uF
f || =√√√√
∑11s=1(u
Jf,s − uF
f,s)2
∑11s=1(u
Ff,s)
2, (12)
where
uJf (r + r) = u f (r) + ∇u f · r (13)
is the Taylor expansion of the fluid velocity around the (ellipsoidal) particle, truncated after the leading-orderterm as done by [18], and uF
f is the exact fluid velocity obtained from the DNS database. On the right hand
Application limits of Jeffery’s theory
10-1 100 10110-5
10-4
10-3
10-2
10-1
100
St=1, uSt=5St=25St=1, vSt=5St=25St=1, wSt=5St=25
10-1 100 10110-5
10-4
10-3
10-2
10-1
100
St=1, du/dzSt=5St=25St=1, dv/dzSt=5St=25St=1, dw/dzSt=5St=25
l+
η+K
||uJ f,i−
uF f,i||
||Ga
,ij−
Gc,i
j||
l+
η+K
St
St
(a) (b)
Fig. 3 Rms error (norm) for fluid velocities (a) and velocity gradients along the z′-axis in the particle frame (b) at varying particlelength, normalized by the reference Kolmogorov length scale η+
k = 2
side of Eq. (12), uJf and uF
f are averaged over the particle length using their discrete values at the uniformlydistributed locations visualized in the top-right inset of Fig. 1, indicated by the subscript s in the summation.The second norm is defined as
||G′a − G
′c|| = |G′
a − G′c|√
2[(G
′a)
2 + (G′c)
2] , (14)
where G′a is the average value of the velocity gradient tensor along the particle (the averaging procedure is the
same adopted for the velocity), and G′c is the velocity gradient tensor evaluated at the particle center of mass.
Both gradients are computed in the particle frame (denoted by the superscript ′); hence, the norm provides anestimate of the root-mean-square (rms) error associated with the use of Eq. (5) to solve for Eq. (5).
The behavior of these two norms is shown in Fig. 3, for the different velocity components (||uJf,i − uF
f,i ||,Fig. 3a) and for the velocity gradients along the z′-axis in the particle frame (||G ′
a,i z − G ′c,i z ||, Fig. 3b). As
expected, a systematic increase of both ||uJf,i − uF
f,i || and ||G ′a,i z − G ′
c,i z || with particle length is observed.
For the velocity norm (Fig. 3a), while following the same qualitative trend with l+/η+K , the difference is
smallest for the streamwise component and significantly larger for the spanwise and wall-normal components.This indicates that the error associated with the streamwise velocity is nearly negligible compared to the errorassociatedwith the spanwise andwall-normal velocities. Comparing particles with different inertia, we observethat short particles (l+/η+
K ≤ 1) with high inertia experience larger differences in velocity approximation thanlow-inertia particles. The norm of the wall-normal velocity gradients (Fig. 3b) reaches a value of about 5 ·10−3
when the particle length equals the Kolmogorov length scale: This means that an error below 1% error isexpected in particle tracking. Overall, all nine components of the gradient tensor yield values of the rms errorwith the same order of magnitude, the largest being observed for the wall-normal components. This findingis in qualitative agreement with the fact that the largest velocity variation occurs along the major axis of theparticle when it is oriented perpendicular to the mean flow.
4.2 Assessment of the applicability limit of the Jeffery formulation for torques
To assess the applicability limit of Eq. (5) we consider again two different norms. The first norm is expressedas
||MaJ − Mc
J || = |MaJ − Mc
J |√
2[(Ma
J )2 + (Mc
J )2] , (15)
J. Ravnik et al.
Fig. 4 Schematic of rod-like particle with cylindrical shape. To compute averaged quantities along the particle length, the particlesurface is discretized into equally spaced sections of length zi
where MaJ is the torque calculated using the velocity gradients averaged along the particle length and Mc
J isthe torque calculated using only the velocity gradients at the particle center of mass. In this case, the particleis modeled as a prolate ellipsoid.
The second norm is obtained extending the assessment to rod-like particles, modeled as cylinders of length2b and cross-section diameter 2a. For these particles, it is easy to compute the torque by discretizing thecylinder into equally sized sections along the major axis (from end to end), calculating for each section thecontribution to the total torque associated with the velocity gradients at the section’s center of mass. Let z jbe the length of the j-th section and let G′
i be the velocity gradient measured at the center of mass of the j-thsection (see Fig. 4). Then, the torque generated by the flow around the j-th section can be written as
M j = μ
∫
� j
r × G′j · nd�, (16)
where � j is the area of the section, r is the location vector pointing from the cylinder’s center to the surface ofthe section, and n is the unit vector normal to the cylinder surface. The fluid velocity gradient at the section’scenter is computed upon trilinear interpolation and is assumed constant over the lateral surface of the section:this way, integration of Eq. (16) is purely geometrical.
Taking into account the moments of inertia of a cylinder:
Iz′ = πλa5ρp, Ix ′ = Iy′ = πλa5ρp3 + 4λ2
6, (17)
the torques can be written as follows (in wall units):
⎛
⎝Mx
i /Ix ′My
i /Iy′Mz
i /Iz′
⎞
⎠ = 1
(a+)2
ρ f
ρp
⎡
⎢⎣
⎛
⎜⎝
63+4λ2
(− f ′ + (ξ ′ − ωx ′))6
3+4λ2(g′ + (η′ − ωy′))
0
⎞
⎟⎠
z′=−b
+
+n∑
i=1
z′+iλa+
⎛
⎜⎝
63+4λ2
( f ′ + (ξ ′ − ωx ′))6
3+4λ2(−g′ + (η′ − ωy′))2(χ ′ − ωz′)
⎞
⎟⎠
z′=z′i
+⎛
⎜⎝
63+4λ2
(− f ′ + (ξ ′ − ωx ′))6
3+4λ2(g′ + (η′ − ωy′))
0
⎞
⎟⎠
z′=+b
⎤
⎥⎥⎦ .
(18)
From Eq. (18), the torqueMcc, computed using the velocity gradients at the particle center of mass, and the
torqueMmc , computed using the velocity gradients averaged along the particle length, can be obtained and the
Application limits of Jeffery’s theory
10-1 100 10110-5
10-4
10-3
10-2
10-1
100
St=1, x’, y’ St=5St=25St=1, z’St=5St=25
10-1 100 10110-5
10-4
10-3
10-2
10-1
100
St=1, x’, y’ St=5St=25St=1, z’St=5St=25
l+
η+K
l+
η+K
||Ma J,i−
Mc J,i||
||Mm c,i−
Mc c,i||
St St
(b)(a)
Fig. 5 Rms error (norm) for hydrodynamic torques at varying particle length, normalized by the reference Kolmogorov lengthscale η+
k = 2. Panels: a rms error for torques on a prolate ellipsoid; b rms error for torques on a cylinder
norm ||Mmc − Mc
c|| calculated. The behavior of the components of ||MaJ − Mc
J || and ||Mmc − Mc
c|| at varyingparticle length and inertia is shown in Fig. 5: Specifically, the x ′- and y′-components (which are identical) arerepresented by the filled squares, and the z′-component is represented by the filled circles. The norms increasewith length reaching an asymptotic value for very long particles (l+/η+
K > 10 in our simulations). The rmserror can be as high as 60 % since these particles extend across regions of flow where the velocity gradientschange significantly, thus violating the conditions for which the Jeffery formulation is valid (regardless oftheir orientation). As far as particle inertia is concerned, we observe that the rms error decreases as the Stokesnumber increases. Therefore, the results provided by the Jeffery formulation are more accurate for particleswith higher inertia. This can be explained considering that particles exhibit preferential concentration andpreferential orientation. In the present flow configuration, the highest degree of preferential concentration andwall accumulation is observed for the St = 25 particles, which are also those with the strongest tendency toalign in the streamwise (wall-parallel) direction. A similar trend was observed by Mortensen et al. [24,25]and by [20] for the same flow configuration. When particles are preferentially aligned with the streamwisedirection, the fluid velocity gradients measured along the particle are subject to smaller variations compared toother orientations. This is especially true in the near-wall region, where the predominant velocity gradient isthe one associated with the mean shear, ∂u/∂z: Because of preferential orientation, the St = 25 particles aremore likely to sample a nearly constant value of ∂u/∂z, which is exactly the condition required for Jeffery’stheory to be valid. Comparing the z′-components of the torque at varying St (filled circles), we find that anincrease in particle inertia results in lower values of the norm. In addition, Fig. 5 also shows that inertia has aweaker effects on the norm along directions perpendicular to the particle major axis (filled squares).
Results shown in Fig. 5 demonstrate clearly that the rms error exhibits a power-law behavior for particleswith l+/η+
K ≤ 8, followed by a plateau for longer particles. The simulation data in these two regions of theplot, below and above l+/η+
K � 8, can be approximated by a fit of the form α(l+/η+K )β . The coefficients α and
β providing the best fit are given in Table 2, while the resulting trends are shown in Fig. 6. In this figure, onlythe results relative to the z′-component of the torques are shown for ease of discussion. The particle length atwhich the power-law fits intersect is denoted by l+0 and provides a threshold value for the range of validity ofthe tabulated values for α and β. The expression α(l+/η+
K )β can be further modified to incorporate the Stokesnumber dependence of the rms error incurred in the calculation of the Jeffery torques. Based on Table 2, wesuggest the following correlation for the global rms error:
||MaJ − Mc
J || ≈ 0.041St−0.34
(l+
η+K
)1.44
forl+
η+K
< 8. (19)
This correlation may be useful to estimate a priori the expected error associated with Lagrangian tracking oncethe physical and geometrical parameters of the particle are known. Note that, according to Eq. (19), an rms
J. Ravnik et al.
Table 2 Power-law fitting of the rms error quantified the ||Mmc − Mc
c|| normSt l+ < l+0 l+ > l+0 l+0 /η+
K
α β α β
Wall-normal component (z′)1 0.03738 1.4335 0.5506 0.0619 7.15 0.02502 1.4404 0.3572 0.1619 8.025 0.01216 1.4844 0.1650 0.3405 9.8
Streamwise (x ′) and spanwise (y′) components1 0.04429 1.4127 0.5571 0.0562 6.55 0.02503 1.4403 0.3567 0.1623 8.025 0.01216 1.4842 0.1650 0.3407 9.8
Simulation data are fitted by a law of the form α(l+/η+K )β . Values of the fitting coefficients α and β are given considering two
different ranges: one for ellipsoidal particles with l+/η+K < 8 (for which the rms error increases significantly with length) and
one for ellipsoidal particles with l+/η+K > 8, for which the error tends toward an asymptotic value. The dimensionless length at
which the prediction of the two fits is the same is given by l+0
10-1 100 10110-4
10-3
10-2
10-1
100
St=1St=5St=25
||Ma J,z
−M
c J,z||
l+0 /η+K
St
l+
η+K
Fig. 6 Power-law fitting of the wall-normal component of Jeffery torques, corresponding to the coefficients in Table 2
error of about 4% should be expected for the Jeffery torques at St = 1 and l+/η+K � 1, this error being much
smaller at higher Stokes numbers.To conclude our analysis, we examine the behavior of the rms error along the wall-normal coordinate. To
this aim, we divided the channel into equally spaced, wall-parallel bins and averaged the torque componentsover all particles within each bin. The values so obtained were then averaged in time. The results are shownin Fig. 7. We observe that the rms error does not depend much on wall distance in the case of particles muchlonger than the Kolmogorov length scale. For shorter particles, we observe a smaller rms error for particleslocated in the central region of the channel (especially for those with low inertia in the top panels of Fig. 7)and for high-inertia particles in the near-wall region (bottom panels of Fig. 7). When moving close to the wall,long-enough particles are preferentially aligned in a plane parallel to the wall and do not experience the largestvelocity gradients, which occur in the wall-normal direction: Accordingly, we observe a decrease of the rmserror. On the other hand, low-inertia particles exhibit a low rms error in the center of the channel, where nopreferential alignment is attained [20].
5 Conclusions
In this paper, we provide a statistical assessment of the application limits of the Jeffery formulation forhydrodynamic torques. The assessment is based on accurate DNS-based Eulerian–Lagrangian datasets for thereference case of prolate ellipsoidal particles and rod-like particles in turbulent channel flow. These elongatedparticles are typically used to model the behavior of rigid fiber suspensions in fluid flows. When their length islarger than the Kolmogorov length scale of turbulence, the flow field around the particles undergoes significant
Application limits of Jeffery’s theory
z+
rela
tive
RM
S
10-1 100 101 102
10-4
10-3
10-2
10-1
100
101
0.10.361.083.67.21836
10-1 100 101 102
10-4
10-3
10-2
10-1
100
10-1 100 101 102
10-4
10-3
10-2
10-1
100
z+
rela
tive
RM
S10-1 100 101 102
10-4
10-3
10-2
10-1
100
101
0.10.361.083.67.21836
10-1 100 101 102
10-4
10-3
10-2
10-1
100
10-1 100 101 102
10-4
10-3
10-2
10-1
100
||Ma J.i−
Mc J.i||
||Ma J.i−
Mc J.i||
z -componentx -component
ll
l l
z+
z+z+
z+
lηk
St = 1
St = 25
Fig. 7 Rms error (norm) for hydrodynamic torques on a prolate ellipsoidal particles as a function of the wall-normal coordinate.Top (resp. bottom) panels refer to the St = 1 (resp. St = 25) particles. Left-hand (resp. right-hand) panels refer to the rms errorassociated with the calculation of the streamwise (resp. wall-normal) component Jeffery torque component
spatial variations and cannot be approximated by extrapolation from the values attained by the fluid velocityat the center of mass of the particles. In turn, calculation of the torques based only on the velocity gradientsat the center of mass becomes inaccurate. We have quantified such inaccuracy for particles with different sizeand different inertia. Our results show that the rms error associated with the use of the Jeffery formulationincreases exponentially with particle length. However, the error is still tolerable (a few percents) for lengthsof the order of the Kolmogorov scale and is found to decrease at increasing particle inertia. This suggests thatJeffery torques can grant acceptable accuracy when used to study the rotational dynamics of elongated particlescharacterized by nonzero values of the particle Reynolds number. From a practical point of view, it may beconcluded that Jeffery-based Lagrangian tracking may be safely applied to environmental problems such asplankton dynamics in aquatic turbulence, since the size of planktonic organisms may vary from scales smallerthan the Kolmogorov length of oceanic turbulence to scales lying within the inertial subrange of turbulence[7]. We expect the approach to be also applicable to problems in which particle dynamics is dominated by thelarger and most energetic flow scales. In such a situation, large-eddy simulation (LES) of turbulence can beperformed: In LES, only eddies of size larger than a given cut-off length scale ξ are computed, whereas eddiesof size smaller than ξ are modeled. Typically, the condition ξ > ηK holds, meaning that ξ is the smallestresolved length scale and that velocity gradients are smoothed out by filtering. This implies that the stringentpoint-particle DNS requirement l << ηK relaxes to l << ξ , a condition that can be satisfied also by particlesbigger than the Kolmogorov scale [3,23].
We also show that the magnitude of the error is affected by particle orientation only for particles smallerthan the Kolmogorov scale: In this case, the error is much smaller for particles aligned in the streamwisedirection (a condition that has a significant statistical occurrence in the near-wall region of bounded flows)compared to particles aligned in the wall-normal direction. The error for long particles is not very sensitive totheir orientation. To complete the statistical characterization of the rms error, we also examined its behavioras a function of particle position relative to the wall. The error is found to depend weakly on particle distancefrom the wall for particles with low-inertia fibers, whereas high-inertia particles are characterized by a smallererror when they are located either very close to the wall or in the center of the channel.
J. Ravnik et al.
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